Filter-Menger set of reals in Cohen extensions

We prove that for every ultrafilter $U$ on $\omega$ there exists a filter $F$ on $2^{<\omega }$ which is $U$-Menger and $\chi \left(F\right)=b\left(U\right)$. We show that in the Cohen model there exists such $F$ which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character $d$ that is not Hurewicz in the $\kappa$-Cohen model where $\kappa >{\omega }_1$ is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with $b<d$. We also study the filter $F$ generated by the set of mutually Cohen reals in the $\kappa$-Cohen model. We prove that $b\left(F\right)={\omega }_1$ and $d\left(F\right)=\kappa$ and every ${\le }^{\ast }$-dominating family in the ground model is $F$-unbounded in extension. Two questions are posed.